GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL*

doi:10.1007/s10473-023-0510-z

摘要/Abstract

摘要： In this paper, we study a quantum kinetic-fluid model in a three-dimensional torus. This model is a coupling of the Vlasov-Fokker-Planck equation and the compressible quantum Navier-Stokes equations with degenerate viscosity. We establish a global weak solution to this model for arbitrarily large initial data when the pressure takes the form $ p(\rho)=\rho^\gamma+p_c(\rho)$, where $\gamma>1$ is the adiabatic coefficient and $p_c(\rho)$ satisfies \begin{equation*} p_c(\rho)=\left\{ \begin{array}{ll}-c\rho^{-4k}\; \;\;\;&{\rm{if}}\;\;\rho\leq 1, \\ \rho^{\gamma}\;\;\;\;&{\rm{if}}\;\;\rho>1 \end{array} \right. \end{equation*} for $k\geq 4$ and some constant $c>0$.

关键词: kinetic-fluid model, quantum Bohm potential, density-dependent viscosity, weak solutions

Abstract: In this paper, we study a quantum kinetic-fluid model in a three-dimensional torus. This model is a coupling of the Vlasov-Fokker-Planck equation and the compressible quantum Navier-Stokes equations with degenerate viscosity. We establish a global weak solution to this model for arbitrarily large initial data when the pressure takes the form $ p(\rho)=\rho^\gamma+p_c(\rho)$, where $\gamma>1$ is the adiabatic coefficient and $p_c(\rho)$ satisfies \begin{equation*} p_c(\rho)=\left\{ \begin{array}{ll}-c\rho^{-4k}\; \;\;\;&{\rm{if}}\;\;\rho\leq 1, \\ \rho^{\gamma}\;\;\;\;&{\rm{if}}\;\;\rho>1 \end{array} \right. \end{equation*} for $k\geq 4$ and some constant $c>0$.

Key words: kinetic-fluid model, quantum Bohm potential, density-dependent viscosity, weak solutions

中图分类号:

35Q35

Fucai Li, Yue Li, Baoyan Sun. GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL^*[J]. 数学物理学报(英文版), 2023, 43(5): 2089-2107.

Fucai Li, Yue Li, Baoyan Sun. GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 2089-2107.

参考文献

[1] Baranger C, Boudin L, Jabin P-E, Mancini S. A modeling of biospray for the upper airways. ESAIM Proc, 2005, 14: 41-47
[2] Baranger C, Desvillettes L. Coupling Euler and Vlasov equations in the context of sprays: the local-in-time, classical solutions. J Hyperbolic Differ Equ, 2006, 3(1): 1-26
[3] Berres S Bürger R, Karlsen K H, Tory E M. Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J Appl Math, 2003, 64(1): 41-80
[4] Berres S Bürger R, Tory E M. Mathematical model and numerical simulation of the liquid fluidization of polydisperse solid particle mixtures. Comput Vis Sci, 2004, 6(2/3): 67-74
[5] Bresch D, Desjardins B. On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models. J Math Pures Appl, 2006, 86(4): 362-368
[6] Bresch D, Desjardins B. On the existence of global weak soutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. J Math Pures Appl, 2007, 87(1): 57-90
[7] Bresch D, Desjardins B, Lin C-K. On some compressible fluid models: Korteweg, lubrication, shallow water systems. Comm Partial Differential Equations, 2003, 28(3/4): 843-868
[8] Brull S, Méhats F. Derivation of viscous correction terms for the isothermal quantum Euler model. Z Angew Math Mech, 2010, 90(3): 219-230
[9] Bürger R, Wendland W L, Concha F. Model equations for gravitational sedimentation-consolidation processes. Z Angew Math Mech, 2000, 80(2): 79-92
[10] Cao W, Jiang P. Global bounded weak entropy solutions to the Euler-Vlasov equations in fluid-particle system. SIAM J Math Anal, 2021, 53(4): 3958-3984
[11] Carrillo J A, Goudon T. Stability and asymptotic analysis of a fluid-particle interaction model. Comm Partial Differential Equations, 2006, 31(7/9): 1349-1379
[12] Chae M, Kang K, Lee J. Global classical solutions for a compressible fluid-particle interaction model. J Hyperbolic Differ Equ, 2013, 10(3): 537-562
[13] Choi Y-P, Jung J. Asymptotic analysis for a Vlasov-Fokker-Planck/Navier-Stokes system in a bounded domain. Math Models Methods Appl Sci, 2021, 31(11): 2213-2295
[14] Choi Y-P, Jung J. Asymptotic analysis for Vlasov-Fokker-Planck/compressible Navier-Stokes equations with a density-dependent viscosity. AIMS Ser Appl Math, 2020, 10: 145-163
[15] Duan R, Liu S. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinet Relat Models, 2013, 6(4): 687-700
[16] Gisclon M, Lacroix-Violet I. About the barotropic compressible quantum Navier-Stokes equations. Nonlinear Anal, 2015, 128: 106-121
[17] Jüngel A. Global weak solutions to compressible Navier-Stokes equations for quantum fluids. SIAM J Math Anal, 2010, 42(3): 1025-1045
[18] Jüngel A. Effective velocity in compressible Navier-Stokes equations with third-order derivatives. Nonlinear Anal, 2011, 74(8): 2813-2818
[19] Karper T K, Mellet A, Trivisa K. Existence of weak solutions to kinetic flocking models. SIAM J Math Anal, 2013, 45(1): 215-243
[20] Lacroix-Violet I, Vasseur A. Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit. J Math Pures Appl, 2018, 114(9): 191-210
[21] Li F, Li Y. Global weak solutions for a kinetic-fluid model with local alignment force in a bounded domain. Commun Pure Appl Anal, 2021, 20(10): 3583-3604
[22] Li F, Li Y, Sun B.Global weak solutions and asymptotic analysis for a kinetic-fluid model with a heterogeneous friction force. preprint
[23] Li F, Mu Y, Wang D. Strong solutions to the compressible Navier-Stokes-Vlasov-Fokker-Planck equations: global existence near the equilibrium and large time behavior. SIAM J Math Anal, 2017, 49(2): 984-1026
[24] Li H-L, Shou L-Y. Global well-posedness of one-dimensional compressible Navier-Stokes-Vlasov system. J Differential Equations, 2021, 280: 841-890
[25] Li H-L, Shou L-Y. Global weak solutions for compressible Navier-Stokes-Vlasov-Fokker-Planck system. Commun Math Res, 2023, 39(1): 136-172
[26] Li Y. Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system with nonhomogeneous boundary data. Z Angew Math Phys, 2021, 72(2): Art. 51
[27] Li Y, Sun B. Global weak solutions to a quantum kinetic-fluid model with large initial data. Nonlinear Analysis: Real World Applications, 2023, 71: Art 103822
[28] Mellet A, Vasseur A. Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations. Math Models Methods Appl Sci, 2007, 17(7): 1039-1063
[29] Mellet A, Vasseur A. Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations. Comm Math Phys, 2008, 281(3): 573-596
[30] Mucha P-B Pokorný M, Zatorska E. Chemically reacting mixtures of degenerated parabolic setting. J Math Phys, 2013, 54(7): 311-341
[31] Oron A, Davis S-H, Bankoff S-G. Long-scale evolution of thin liquid films. Rev Mod Phys, 1997, 69: 931-980
[32] Sartory W K. Three-component analysis of blood sedimentation by the method of characteristics. Math Biosci, 1977, 33(1/2): 145-165
[33] Simon J. Compact sets in the space $L^p(0, T;B)$. Ann Math Pure Appl, 1986, 146: 65-96
[34] Spannenberg A, Galvin K P. Continuous differential sedimentation of a binary suspension. Chem Engrg Aust, 1996, 21: 7-11
[35] Tang T, Niu C.Global existence of weak solutions to the quantum Navier-Stokes equations
(in Chinese). Acta Mathematica Scientia, 2022, 42A(2): 387-400
[36] Vasseur A, Yu C. Global weak solutions to the compressible quantum Navier-Stokes equations with damping. SIAM J Math Anal, 2016, 48(2): 1489-1511
[37] Williams F A. Spray combustion and atomization. Physics of Fluids, 1958, 1(6): 541-555

GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL^*

GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL^*

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	Jingpeng, Wu, Xianwen Zhang. THE ENERGY CONSERVATION OF VLASOV-POISSON SYSTEMS^*[J]. 数学物理学报(英文版), 2023, 43(2): 668-674.
[2]	Quansen Jiu, Lin Ma. GLOBAL WEAK SOLUTIONS OF COMPRESSIBLE NAVIER-STOKES-LANDAU-LIFSHITZ-MAXWELL EQUATIONS FOR QUANTUM FLUIDS IN DIMENSION THREE^*[J]. 数学物理学报(英文版), 2023, 43(1): 25-42.
[3]	苏奕帆, 郭真华. ZERO DISSIPATION LIMIT TO RAREFACTION WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SELECTED DENSITY-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2021, 41(5): 1635-1658.
[4]	何一鸣, 訾瑞昭. ENERGY CONSERVATION FOR SOLUTIONS OF INCOMPRESSIBLE VISCOELASTIC FLUIDS[J]. 数学物理学报(英文版), 2021, 41(4): 1287-1301.
[5]	任亚伯, 郭柏灵, 王术. GLOBAL WEAK SOLUTIONS TO THE α-MODEL REGULARIZATION FOR 3D COMPRESSIBLE EULER-POISSON EQUATIONS[J]. 数学物理学报(英文版), 2021, 41(3): 679-702.
[6]	孙庆有, 陆云光, Christian KLINGENBERG. GLOBAL WEAK SOLUTIONS FOR A NONLINEAR HYPERBOLIC SYSTEM[J]. 数学物理学报(英文版), 2020, 40(5): 1185-1194.
[7]	蒲学科, 郭柏灵. INITIAL BOUNDARY VALUE PROBLEM FOR THE 3D MAGNETIC-CURVATURE-DRIVEN RAYLEIGH-TAYLOR MODEL[J]. 数学物理学报(英文版), 2020, 40(2): 529-542.
[8]	郭闪闪, 谭忠. ON INTEGRABILITY UP TO THE BOUNDARY OF THE WEAK SOLUTIONS TO A NON-NEWTONIAN FLUID[J]. 数学物理学报(英文版), 2019, 39(2): 420-428.
[9]	叶嵎林. GLOBAL CLASSICAL SOLUTION TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2016, 36(5): 1419-1432.
[10]	李进开, 辛周平. GLOBAL EXISTENCE OF WEAK SOLUTIONS TO THE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS IN 2D[J]. 数学物理学报(英文版), 2016, 36(4): 973-1014.
[11]	连汝续, 刘健. FREE BOUNDARY VALUE PROBLEM FOR THE CYLINDRICALLY SYMMETRIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2016, 36(1): 111-123.
[12]	何躏, 唐少君, 王涛. STABILITY OF VISCOUS SHOCK WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2016, 36(1): 34-48.
[13]	王伟伟. ON GLOBAL BEHAVIOR OF WEAK SOLUTIONS OF COMPRESSIBLE FLOWS OF NEMATIC LIQUID CRYSTALS[J]. 数学物理学报(英文版), 2015, 35(3): 650-672.
[14]	李为, 汪星, 黄南京. DIFFERENTIAL INVERSE VARIATIONAL INEQUALITIES IN FINITE DIMENSIONAL SPACES[J]. 数学物理学报(英文版), 2015, 35(2): 407-422.
[15]	叶嵎林, 窦昌胜, 酒全森. LOCAL WELL-POSEDNESS TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2014, 34(3): 851-871.